Finite Group Databasebeta

← #7196896

Subgroups — #7196896

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59870C211yesmaximal, characteristic, complemented
2#59822C211yesmaximal, characteristic
3#32669C214yesmaximal, complemented
4#32164C214yesmaximal, complemented
5#28404C212yesmaximal, complemented
6#28202C211yesmaximal, characteristic, complemented
7#28160C211yesmaximal, characteristic, complemented
8#27488C214yesmaximal, complemented
9#26318C214yesmaximal, complemented
10#24680C212yesmaximal, complemented
11#24628C212yesmaximal, complemented
12#23799C211yesmaximal, characteristic, complemented
13#22805C212yesmaximal, complemented
14#22307C211yesmaximal, characteristic, complemented
15#22219C211yesmaximal, characteristic, complemented
16C2×#555—21no—
17C2×#486C2211yescharacteristic
18C2×#476C2212yes—
19C2×#460C2212yes—
20C2×C4×SD16C2212yes—
21C2×C4×SD16C2212yes—
22#2805—22no—
23#2733—22no—
24#2581—22no—
25#2571—21no—
26#2568C2212yes—
27#2567C2212yes—
28#2548C2214yes—
29#2541C2212yes—
30#2541C2212yes—
31#2537—21no—
32#2511C2214yes—
33#2509—22no—
34#2414—22no—
35#2414—22no—
36#2365—21no—
37#2355C2212yes—
38#2354C2212yes—
39#2354C2212yes—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2096C2212yes—
47#2094C2214yes—
48#2094C2212yes—
49#2084C2212yes—
50#2071C2212yes—
51#2071C2214yes—
52#2055C2214yes—
53#2055C2214yes—
54#2041C2212yes—
55#2014C2214yes—
56#2014C2214yes—
57#2014C2212yes—
58#2013C2212yes—
59#2013C2212yes—
60#1990C2211yescharacteristic
61#1984C2214yes—
62#1982C2214yes—
63#1968C2214yes—
64#1968C2214yes—
65#1957C2211yescharacteristic
66#1957C2214yes—
67#1948C2214yes—
68#1939C2212yes—
69#1927C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1926C2214yes—
73#1926C2214yes—
74#1926C2214yes—
75#1916C2214yes—
76#1877C2214yes—
77#1864C2214yes—
78#1844C2211yescharacteristic
79#1831C2211yescharacteristic
80#1803C2212yes—
81#1802C2212yes—
82C4×#106C2214yes—
83C8×C4⋊C4C2212yes—
84#1744C2211yescharacteristic
85C22×SD16C2312yes—
86C22×SD16C2312yes—
87C22×SD16C2314yes—
88#555—48no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#127C2314yes—
95C2×#124—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—22no—
99C2×#121—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—22no—
111C2×#117—22no—
112#486—24no—
113#476—44no—
114#460—44no—
115C4×SD16—28no—
116C4×SD16—44no—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2312yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2314yes—
135C2×C4×C8C2311yescharacteristic
136#400—24no—
137#400—24no—
138#400—24no—
139#395—24no—
140#394—24no—
141#394—24no—
142#392—24no—
143#392—24no—
144#391—22no—
145#391—21no—
146#391D412yes—
147#391C2312yes—
148#391C2314yes—
149#391C2314yes—
150#390—24no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390C2312yes—
156#390C2312yes—
157#390C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389D412yes—
167#389D412yes—
168#389C2312yes—
169#388—24no—
170#388—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#384D411yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2314yes—
180#382—22no—
181#382—22no—
182#380—22no—
183#380—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C43C2311yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2314yes—
201#336C2312yes—
202#336C2312yes—
203C22×Q8—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2412yes—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4C2412yes—
213C22×D4C2412yes—
214C23×C4—22no—
215C23×C4—22no—
216C2×SD16—24no—
217C2×SD16—42no—
218C2×SD16—42no—
219C2×SD16—42no—
220C2×SD16—42no—
221C2×SD16—42no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C22×C8C2412yes—
226C22×C8C2414yes—
227C22×C8C2412yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—42no—
231C4⋊Q8SD1618yescomplemented
232#127—22no—
233#127—22no—
234#127—42no—
235#127—44no—
236#127—44no—
237#124—416no—
238#121—48no—
239#121—48no—
240#121—84no—
241#121—84no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—44no—
245C4×Q8—82no—
246C4×Q8—84no—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—44no—
252C4×D4—82no—
253C4×D4—82no—
254#117—48no—
255#117—48no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304#106—24no—
305#106—44no—
306#106—42no—
307#106—44no—
308C4⋊C8—24no—
309C4⋊C8—44no—
310#103—24no—
311#103—44no—
312#103—42no—
313#102—24no—
314#102—28no—
315#102—44no—
316#102—44no—
317#102—42no—
318#102—44no—
319#102—44no—
320C4×C8—22no—
321C4×C8—44no—
322C4×C8#4114yes—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—22no—
336C24—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8C2×SD1618yes—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4C22×D414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D414yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2511yescharacteristic
398SD16—48no—
399SD16—48no—
400SD16—88no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8C2×#4114yes—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C4⋊C4—84no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—82no—
437C4⋊C4C2×SD1618yes—
438C4⋊C4C2×SD1618yes—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—44no—
457C42—44no—
458C42—48no—
459C42—84no—
460C42C2×SD1614yes—
461C42C2×#4112yes—
462C42H(5,𝔽2)12yes—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C23×D411yescharacteristic
478Q8—48no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—84no—
484D4—42no—
485D4—44no—
486D4—42no—
487D4—42no—
488D4—44no—
489D4—42no—
490D4—84no—
491D4—84no—
492D4—84no—
493D4—84no—
494D4—168no—
495D4—168no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—88no—
532C2×C4—88no—
533C2×C4#55012yes—
534C2×C4#55012yes—
535C2×C4C22×SD1618yes—
536C2×C4C22×SD1612yes—
537C2×C4C22×SD1612yes—
538C2×C4C22×SD1612yes—
539C2×C4#57712yes—
540C2×C4C22×#4111yescharacteristic
541C2×C4C2×H(5,𝔽2)11yescharacteristic
542C2×C4#58512yes—
543C2×C4#58512yes—
544C8—24no—
545C8—42no—
546C8—44no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×#55011yescharacteristic
562C22#348511yescharacteristic
563C22#348711yescharacteristic
564C22C23×SD1611yescharacteristic
565C22C2×#57711yescharacteristic
566C22#358511yescharacteristic
567C22#358511yescharacteristic
568C4—28no—
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—44no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4#339214yes—
582C4#349312yes—
583C4#354012yes—
584C2—162no—
585C2—162no—
586C2#4538811yescharacteristic
587C2C2×#339211yescharacteristic
588C2#6141411yescharacteristic
589C2#6169711yescharacteristic
590C2#6180511yescharacteristic
591C2#6188111yescharacteristic
592C2#6191311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59822C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #59870C2semidirect
3. #32669C2semidirect
4. #32164C2semidirect
5. #28404C2semidirect
6. #28202C2semidirect
7. #28160C2semidirect
8. #27488C2semidirect
9. #26318C2semidirect
10. #24680C2semidirect
11. #24628C2semidirect
12. #23799C2semidirect
13. #22805C2semidirect
14. #22307C2semidirect
15. #22219C2semidirect
231. C4⋊Q8SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#598702
#598222
#326692
#321642
#284042
#282022
#281602
#274882
#263182
#246802
#246282
#237992
#228052
#223072
#222192